Heat Rash / Magic Merlin Sleepsuit

My baby has started using a magic Merlin sleepsuit and she absolutely loves it. She pretty much won’t sleep unless she’s wearing it or napping in my arms. We keep the room cool with the fan on, and one thin short sleeve onesie underneath her suit - but she’s developing a heat rash. Does anyone else have this problem? Or have a sleep solution we could switch to? I don’t want her to overheat!